Machine Learning Summer School 2007 - Tuebingen

Convex Optimization

author: Lieven Vandenberghe, University of California Los Angeles

Description

The lectures will provide an introduction to the theory and applications of convex optimization. The emphasis will be on results useful for convex modeling, i.e., recognizing and formulating convex optimization problems in practice. * The first lecture will introduce some of the fundamental theory of convex sets and functions. * In lecture 2 we will discuss general properties of convex optimization problems and define important standard classes (linear and quadratic programming, second-order cone programming, semidefinite programming) and their applications. * In lecture 3 the material will be illustrated with various examples.

Categories

Top: Mathematics: Operations Research

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*Slides*
0:00	Convex optimization: Modeling and applications
0:35	Outline
1:18	Lecture 1
1:21	Introduction
1:25	Mathematical optimization
3:33	Least squares
5:34	Linear programming
7:34	Convex optimization problem-part01
8:22	Convex optimization problem-part02
9:56	History
11:57	New applications since 1990
12:52	Interior-point methods
14:13	Convex sets
14:22	Definition
15:21	Convex sets - A
15:48	Hyperplanes and halfspaces
16:31	Euclidean balls and ellipsoids
17:22	Norm balls and norm cones
18:41	Polyhedra
19:25	Positive semidefinite cone
21:04	Operations that preserve convexity
21:24	Intersection
24:39	Affine function
27:07	Norm balls and norm cones - A
27:16	Affine function - A
27:47	Prospective and linear fractional function
28:41	Example
32:21	Convex functions
32:47	Definition
33:40	Examples on R
34:21	Examples R and R
37:09	Differentiable convex functions
39:31	Examples
41:25	Operations that preserve convexity
41:57	Positive weighted sum and composition with affine function
43:26	Pointwise maximum
45:41	Pointwise supremum

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